While the effect might be measurable (just barely) it's small. We add about 30 billion tons of CO2 to the air every year. About half of that gets absorbed into the oceans (give or take), the remaining 15 billion tons remains in the atmosphere for a long time.
280 PPM CO2 the post ice-age, pre industrial revolution number works out to about 1,000 billion tons of CO2. Adding 15 billion tons per year, at the current rate, it would take about 66 years to double the 280 PPM to 560 PPM. That's, of-course, very much an approximation and the ocean absorption rate probably doesn't remain constant it depends on the change in concentration and change in ocean temperature.
Water vapor, however. While the output varies depending on fuel the mass of water added by burning fossil fuels is about 1/2 the mass of CO2, so 15 billion tons per year. The amount of water vapor in the atmosphere at any given time is by this estimate about 5,700 billion tons. Adding 15 billion tons of water vapor to the atmosphere per year might be enough to be significant, but as pointed out in the comments, water vapor leaves the atmosphere very quickly. Unlike CO2, which takes a long time to filter out. The half-life of water is days, the half-life of CO2 is probably a century or more.
If extensive driving or fossil fuel burning is done in low humidity areas, it seems feasible that some warming could be added by the increase in water vapor, but I expect that effect would be small but perhaps measurable. In high humidity areas it would be irrelevant. I've never heard of a study that looked at this in any detail. That said, the intuitively obvious answer is that the effect would be tiny. A tiny fraction of of the effect of CO2. That said, water vapor does need to be included in the study as warmer temperatures hold more water vapor, so, as the Earth warms, water vapor in the atmosphere increases and pushes the warming along even more. That's a well known feedback mechanism. The effect of water from burning fossil fuels is likely very tiny because it's too little and too temporary and the net total amount of water vapor in the air is still primarily a factor of temperature, weather patterns and wind direction. It is, however, probably not zero. There's likely some tiny effect, but I've never heard of a study that looked into it.